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<li><a href="./">The Open Quant Live Book</a></li>

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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a><ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#description"><i class="fa fa-check"></i>Description</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#contribute"><i class="fa fa-check"></i>Contribute</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#working-contents"><i class="fa fa-check"></i>Working Contents</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#books-information"><i class="fa fa-check"></i>Book’s information</a></li>
</ul></li>
<li class="part"><span><b>I The Basics</b></span></li>
<li class="chapter" data-level="1" data-path="io.html"><a href="io.html"><i class="fa fa-check"></i><b>1</b> I/O</a><ul>
<li class="chapter" data-level="1.1" data-path="io.html"><a href="io.html#importing-data"><i class="fa fa-check"></i><b>1.1</b> Importing Data</a><ul>
<li class="chapter" data-level="1.1.1" data-path="io.html"><a href="io.html#text-files"><i class="fa fa-check"></i><b>1.1.1</b> Text Files</a></li>
<li class="chapter" data-level="1.1.2" data-path="io.html"><a href="io.html#excel-files"><i class="fa fa-check"></i><b>1.1.2</b> Excel Files</a></li>
<li class="chapter" data-level="1.1.3" data-path="io.html"><a href="io.html#json-files"><i class="fa fa-check"></i><b>1.1.3</b> JSON Files</a></li>
<li class="chapter" data-level="1.1.4" data-path="io.html"><a href="io.html#large-files"><i class="fa fa-check"></i><b>1.1.4</b> Large Files</a></li>
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<li class="chapter" data-level="1.2" data-path="io.html"><a href="io.html#data-sources"><i class="fa fa-check"></i><b>1.2</b> Data Sources</a><ul>
<li class="chapter" data-level="1.2.1" data-path="io.html"><a href="io.html#alpha-vantage"><i class="fa fa-check"></i><b>1.2.1</b> Alpha Vantage</a></li>
<li class="chapter" data-level="1.2.2" data-path="io.html"><a href="io.html#iex"><i class="fa fa-check"></i><b>1.2.2</b> IEX</a></li>
<li class="chapter" data-level="1.2.3" data-path="io.html"><a href="io.html#quandl"><i class="fa fa-check"></i><b>1.2.3</b> Quandl</a></li>
<li class="chapter" data-level="1.2.4" data-path="io.html"><a href="io.html#sec"><i class="fa fa-check"></i><b>1.2.4</b> SEC</a></li>
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<li class="chapter" data-level="1.3" data-path="io.html"><a href="io.html#conclusion"><i class="fa fa-check"></i><b>1.3</b> Conclusion</a><ul>
<li class="chapter" data-level="1.3.1" data-path="io.html"><a href="io.html#further-reading"><i class="fa fa-check"></i><b>1.3.1</b> Further Reading</a></li>
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<li class="chapter" data-level="2" data-path="stylized-facts.html"><a href="stylized-facts.html"><i class="fa fa-check"></i><b>2</b> Stylized Facts</a><ul>
<li class="chapter" data-level="2.1" data-path="stylized-facts.html"><a href="stylized-facts.html#introduction"><i class="fa fa-check"></i><b>2.1</b> Introduction</a></li>
<li class="chapter" data-level="2.2" data-path="stylized-facts.html"><a href="stylized-facts.html#distribution-of-returns"><i class="fa fa-check"></i><b>2.2</b> Distribution of Returns</a><ul>
<li class="chapter" data-level="2.2.1" data-path="stylized-facts.html"><a href="stylized-facts.html#fat-tails"><i class="fa fa-check"></i><b>2.2.1</b> Fat Tails</a></li>
<li class="chapter" data-level="2.2.2" data-path="stylized-facts.html"><a href="stylized-facts.html#skewness"><i class="fa fa-check"></i><b>2.2.2</b> Skewness</a></li>
</ul></li>
<li class="chapter" data-level="2.3" data-path="stylized-facts.html"><a href="stylized-facts.html#volatility"><i class="fa fa-check"></i><b>2.3</b> Volatility</a><ul>
<li class="chapter" data-level="2.3.1" data-path="stylized-facts.html"><a href="stylized-facts.html#time-invariance"><i class="fa fa-check"></i><b>2.3.1</b> Time-invariance</a></li>
<li class="chapter" data-level="2.3.2" data-path="stylized-facts.html"><a href="stylized-facts.html#volatility-clustering"><i class="fa fa-check"></i><b>2.3.2</b> Volatility Clustering</a></li>
<li class="chapter" data-level="2.3.3" data-path="stylized-facts.html"><a href="stylized-facts.html#correlation-with-trading-volume"><i class="fa fa-check"></i><b>2.3.3</b> Correlation with Trading Volume</a></li>
</ul></li>
<li class="chapter" data-level="2.4" data-path="stylized-facts.html"><a href="stylized-facts.html#correlation"><i class="fa fa-check"></i><b>2.4</b> Correlation</a><ul>
<li class="chapter" data-level="2.4.1" data-path="stylized-facts.html"><a href="stylized-facts.html#time-invariance-1"><i class="fa fa-check"></i><b>2.4.1</b> Time-invariance</a></li>
<li class="chapter" data-level="2.4.2" data-path="stylized-facts.html"><a href="stylized-facts.html#auto-correlation"><i class="fa fa-check"></i><b>2.4.2</b> Auto-correlation</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>II Algo Trading</b></span></li>
<li class="part"><span><b>III Portfolio Optimization</b></span></li>
<li class="chapter" data-level="3" data-path="risk-parity-portfolios.html"><a href="risk-parity-portfolios.html"><i class="fa fa-check"></i><b>3</b> Risk Parity Portfolios</a><ul>
<li class="chapter" data-level="3.1" data-path="risk-parity-portfolios.html"><a href="risk-parity-portfolios.html#introduction-1"><i class="fa fa-check"></i><b>3.1</b> Introduction</a></li>
<li class="chapter" data-level="3.2" data-path="risk-parity-portfolios.html"><a href="risk-parity-portfolios.html#risk-parity-portfolio"><i class="fa fa-check"></i><b>3.2</b> Risk Parity Portfolio</a></li>
<li class="chapter" data-level="3.3" data-path="risk-parity-portfolios.html"><a href="risk-parity-portfolios.html#tangency-portfolio"><i class="fa fa-check"></i><b>3.3</b> Tangency Portfolio</a></li>
<li class="chapter" data-level="3.4" data-path="risk-parity-portfolios.html"><a href="risk-parity-portfolios.html#optimizing-faang-ray-dalio-versus-markowitz"><i class="fa fa-check"></i><b>3.4</b> Optimizing FAANG: Ray Dalio versus Markowitz</a><ul>
<li class="chapter" data-level="3.4.1" data-path="risk-parity-portfolios.html"><a href="risk-parity-portfolios.html#single-portfolio"><i class="fa fa-check"></i><b>3.4.1</b> Single Portfolio</a></li>
<li class="chapter" data-level="3.4.2" data-path="risk-parity-portfolios.html"><a href="risk-parity-portfolios.html#the-ray-dalio-faang-index"><i class="fa fa-check"></i><b>3.4.2</b> The Ray Dalio FAANG Index</a></li>
</ul></li>
<li class="chapter" data-level="3.5" data-path="risk-parity-portfolios.html"><a href="risk-parity-portfolios.html#discussion-and-conclusion"><i class="fa fa-check"></i><b>3.5</b> Discussion and Conclusion</a></li>
</ul></li>
<li class="part"><span><b>IV Machine Learning</b></span></li>
<li class="part"><span><b>V Econophysics</b></span></li>
<li class="chapter" data-level="4" data-path="entropy.html"><a href="entropy.html"><i class="fa fa-check"></i><b>4</b> Entropy</a><ul>
<li class="chapter" data-level="4.1" data-path="entropy.html"><a href="entropy.html#introduction-2"><i class="fa fa-check"></i><b>4.1</b> Introduction</a></li>
<li class="chapter" data-level="4.2" data-path="entropy.html"><a href="entropy.html#nonlinear-coupling"><i class="fa fa-check"></i><b>4.2</b> Nonlinear Coupling</a><ul>
<li class="chapter" data-level="4.2.1" data-path="entropy.html"><a href="entropy.html#simulated-systems"><i class="fa fa-check"></i><b>4.2.1</b> Simulated Systems</a></li>
<li class="chapter" data-level="4.2.2" data-path="entropy.html"><a href="entropy.html#equity-commodities-relationship"><i class="fa fa-check"></i><b>4.2.2</b> Equity-Commodities Relationship</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="entropy.html"><a href="entropy.html#efficiency-and-bubbles-a-case-study-in-the-crypto-and-equity-markets"><i class="fa fa-check"></i><b>4.3</b> Efficiency and Bubbles: A Case Study in the Crypto and Equity Markets</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><i class="fa fa-check"></i><b>5</b> How to Measure Statistical Causality: A Transfer Entropy Approach with Financial Applications</a><ul>
<li class="chapter" data-level="5.1" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#LinearG"><i class="fa fa-check"></i><b>5.1</b> A First Definition of Causality</a></li>
<li class="chapter" data-level="5.2" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#a-probabilistic-based-definition"><i class="fa fa-check"></i><b>5.2</b> A Probabilistic-Based Definition</a></li>
<li class="chapter" data-level="5.3" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#nonlinearG"><i class="fa fa-check"></i><b>5.3</b> Transfer Entropy and Statistical Causality</a></li>
<li class="chapter" data-level="5.4" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#net-information-flow"><i class="fa fa-check"></i><b>5.4</b> Net Information Flow</a></li>
<li class="chapter" data-level="5.5" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#the-link-between-granger-causality-and-transfer-entropy"><i class="fa fa-check"></i><b>5.5</b> The Link Between Granger-causality and Transfer Entropy</a></li>
<li class="chapter" data-level="5.6" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#information-flow-on-simulated-systems"><i class="fa fa-check"></i><b>5.6</b> Information Flow on Simulated Systems</a></li>
<li class="chapter" data-level="5.7" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#information-flow-among-international-stock-market-indices"><i class="fa fa-check"></i><b>5.7</b> Information Flow among International Stock Market Indices</a></li>
<li class="chapter" data-level="5.8" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#other-applications"><i class="fa fa-check"></i><b>5.8</b> Other Applications</a><ul>
<li class="chapter" data-level="5.8.1" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#quantifying-information-flow-between-social-media-and-the-stock-market"><i class="fa fa-check"></i><b>5.8.1</b> Quantifying Information Flow Between Social Media and the Stock Market</a></li>
<li class="chapter" data-level="5.8.2" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#detecting-causal-links-between-investor-sentiment-and-cryptocurrency-prices"><i class="fa fa-check"></i><b>5.8.2</b> Detecting Causal Links Between Investor Sentiment and Cryptocurrency Prices</a></li>
</ul></li>
<li class="chapter" data-level="5.9" data-path="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html"><a href="how-to-measure-statistical-causality-a-transfer-entropy-approach-with-financial-applications.html#conclusions"><i class="fa fa-check"></i><b>5.9</b> Conclusions</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="financial-networks.html"><a href="financial-networks.html"><i class="fa fa-check"></i><b>6</b> Financial Networks</a><ul>
<li class="chapter" data-level="6.1" data-path="financial-networks.html"><a href="financial-networks.html#introduction-3"><i class="fa fa-check"></i><b>6.1</b> Introduction</a></li>
<li class="chapter" data-level="6.2" data-path="financial-networks.html"><a href="financial-networks.html#network-construction"><i class="fa fa-check"></i><b>6.2</b> Network Construction</a><ul>
<li class="chapter" data-level="6.2.1" data-path="financial-networks.html"><a href="financial-networks.html#network-filtering-asset-graphs"><i class="fa fa-check"></i><b>6.2.1</b> Network Filtering: Asset Graphs</a></li>
<li class="chapter" data-level="6.2.2" data-path="financial-networks.html"><a href="financial-networks.html#network-filtering-mst"><i class="fa fa-check"></i><b>6.2.2</b> Network Filtering: MST</a></li>
<li class="chapter" data-level="6.2.3" data-path="financial-networks.html"><a href="financial-networks.html#network-filtering-pmfg"><i class="fa fa-check"></i><b>6.2.3</b> Network Filtering: PMFG</a></li>
</ul></li>
<li class="chapter" data-level="6.3" data-path="financial-networks.html"><a href="financial-networks.html#applications"><i class="fa fa-check"></i><b>6.3</b> Applications</a><ul>
<li class="chapter" data-level="6.3.1" data-path="financial-networks.html"><a href="financial-networks.html#industry-taxonomy"><i class="fa fa-check"></i><b>6.3.1</b> Industry Taxonomy</a></li>
<li class="chapter" data-level="6.3.2" data-path="financial-networks.html"><a href="financial-networks.html#portfolio-construction"><i class="fa fa-check"></i><b>6.3.2</b> Portfolio Construction</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>VI Alternative Data</b></span></li>
<li class="chapter" data-level="7" data-path="the-market-the-players-and-the-rules.html"><a href="the-market-the-players-and-the-rules.html"><i class="fa fa-check"></i><b>7</b> The Market, The Players and The Rules</a><ul>
<li class="chapter" data-level="7.1" data-path="the-market-the-players-and-the-rules.html"><a href="the-market-the-players-and-the-rules.html#the-market"><i class="fa fa-check"></i><b>7.1</b> The Market</a></li>
<li class="chapter" data-level="7.2" data-path="the-market-the-players-and-the-rules.html"><a href="the-market-the-players-and-the-rules.html#the-data"><i class="fa fa-check"></i><b>7.2</b> The Data</a></li>
<li class="chapter" data-level="7.3" data-path="the-market-the-players-and-the-rules.html"><a href="the-market-the-players-and-the-rules.html#the-buyers"><i class="fa fa-check"></i><b>7.3</b> The Buyers</a></li>
<li class="chapter" data-level="7.4" data-path="the-market-the-players-and-the-rules.html"><a href="the-market-the-players-and-the-rules.html#conclusion-1"><i class="fa fa-check"></i><b>7.4</b> Conclusion</a></li>
</ul></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="statistical-methods.html"><a href="statistical-methods.html"><i class="fa fa-check"></i><b>A</b> Statistical Methods</a><ul>
<li class="chapter" data-level="A.1" data-path="statistical-methods.html"><a href="statistical-methods.html#kde"><i class="fa fa-check"></i><b>A.1</b> Kernel Density Estimation</a></li>
</ul></li>
<li class="chapter" data-level="B" data-path="datasets.html"><a href="datasets.html"><i class="fa fa-check"></i><b>B</b> Datasets</a><ul>
<li class="chapter" data-level="B.1" data-path="datasets.html"><a href="datasets.html#dt-indices"><i class="fa fa-check"></i><b>B.1</b> Log-Returns of International Stock Market Indices Prices</a><ul>
<li class="chapter" data-level="B.1.1" data-path="datasets.html"><a href="datasets.html#dataset-location"><i class="fa fa-check"></i><b>B.1.1</b> Dataset Location</a></li>
<li class="chapter" data-level="B.1.2" data-path="datasets.html"><a href="datasets.html#dataset-description"><i class="fa fa-check"></i><b>B.1.2</b> Dataset Description</a></li>
<li class="chapter" data-level="B.1.3" data-path="datasets.html"><a href="datasets.html#data-source"><i class="fa fa-check"></i><b>B.1.3</b> Data Source</a></li>
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<div id="financial-networks" class="section level1">
<h1><span class="header-section-number">Chapter 6</span> Financial Networks</h1>
<div id="introduction-3" class="section level2">
<h2><span class="header-section-number">6.1</span> Introduction</h2>
<p>Financial markets can be regarded as a complex network in which nodes represent different financial assets and edges represent one or many types of relationships among those assets. Filtered correlation-based networks have successfully been used in the literature to study financial markets structure particularly from observational data derived from empirical financial time series <span class="citation">(Bardoscia et al. <a href="#ref-bardoscia2017pathways">2017</a>; S. A. L. Tumminello Michele AND Miccichè <a href="#ref-10.1371/journal.pone.0017994">2011</a>; R. N. Mantegna <a href="#ref-Mantegna1999">1999</a>; T. Aste, Shaw, and Di Matteo <a href="#ref-aste2010correlation">2010</a>; Michele Tumminello, Lillo, and Mantegna <a href="#ref-Tumminello201040">2010</a>, <span class="citation">M. Tumminello et al. (<a href="#ref-Tumminello26072005">2005</a>)</span>)</span>.</p>
<p>The underlying principle is to use correlations from empirical financial time series to construct a sparse network representing the most relevant connections. Analyses on filtered correlation-based networks for information extraction <span class="citation">(Song, Aste, and Di Matteo <a href="#ref-song2008analysis">2008</a>; T. Aste, Shaw, and Di Matteo <a href="#ref-aste2010correlation">2010</a>)</span> have widely been used to explain market interconnectedness from high-dimensional data. Applications include asset allocation <span class="citation">(Y. Li et al. <a href="#ref-LI2018">2018</a>; Pozzi, Di Matteo, and Aste <a href="#ref-pozzi2013spread">2013</a>)</span>, market stability assessments <span class="citation">(Morales et al. <a href="#ref-morales2012dynamical">2012</a>)</span>, hierarchical structure analyses <span class="citation">(R. N. Mantegna <a href="#ref-Mantegna1999">1999</a>; T. Aste, Shaw, and Di Matteo <a href="#ref-aste2010correlation">2010</a>; Michele Tumminello, Lillo, and Mantegna <a href="#ref-Tumminello201040">2010</a>; Musmeci, Aste, and Matteo <a href="#ref-musmeci2014clustering">2014</a>; Song, Di Matteo, and Aste <a href="#ref-song2012hierarchical">2012</a>)</span> and the identification of lead-lag relationships <span class="citation">(Curme, Stanley, and Vodenska <a href="#ref-curme2015coupled">2015</a>)</span>.</p>
<p>In this Chapter we will describe how to</p>
<ul>
<li>Construct and filter financial networks;</li>
<li>Build price-based dynamic industry taxonomies;</li>
<li>Implement a trading strategy based on financial network structure.</li>
</ul>
</div>
<div id="network-construction" class="section level2">
<h2><span class="header-section-number">6.2</span> Network Construction</h2>
<p>We selected <span class="math inline">\(N = 100\)</span> of the most capitalized companies that were part of the S&amp;P500 index from 09/05/2012 to 08/25/2017. The list of these companies’ ticker symbols is reported in the Appendix . For each stock <span class="math inline">\(i\)</span> the financial variable was defined as the daily stock’s log-return <span class="math inline">\(R_i(\tau)\)</span> at time <span class="math inline">\(\tau\)</span>.</p>
<p>Stock returns <span class="math inline">\(R_i\)</span> and social media opinion scores <span class="math inline">\(O_i\)</span> each amounted to a time series of length equals to 1251 trading days. These series were divided time-wise into <span class="math inline">\(M = 225\)</span> windows <span class="math inline">\(t = 1, 2, \ldots, M\)</span> of width <span class="math inline">\(T = 126\)</span> trading days. A window step length parameter of <span class="math inline">\(\delta T = 5\)</span> trading days defined the displacement of the window, i.e., the number of trading days between two consecutive windows. The choice of window width <span class="math inline">\(T\)</span> and window step <span class="math inline">\(\delta T\)</span> is arbitrary, and it is a trade-off between having analysis that is either too dynamic or too smooth. The smaller the window width and the larger the window steps, the more dynamic the data are.</p>
To characterize the synchronous time evolution of assets, we used equal time Kendall’s rank coefficients between assets <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span>, defined as
<span class="math display">\[\begin{equation}
 \rho_{i, j}(t) = \sum\limits_{t&#39; &lt; \tau}sgn(V_i(t&#39;) - V_i(\tau))sgn(V_j(t&#39;) - V_j(\tau)),
\end{equation}\]</span>
<p>where <span class="math inline">\(t&#39;\)</span> and <span class="math inline">\(\tau\)</span> are time indexes within the window <span class="math inline">\(t\)</span> and <span class="math inline">\(V_i \in \{R_i, O_i\}\)</span>.</p>
<p>Kendall’s rank coefficients takes into account possible nonlinear (monotonic) relationships. It fulfill the condition <span class="math inline">\(-1 \leq \rho_{i, j} \leq 1\)</span> and form the <span class="math inline">\(N \times N\)</span> correlation matrix <span class="math inline">\(C(t)\)</span> that served as the basis for the networks constructed in this work. To construct the asset-based financial and social networks, we defined a distance between a pair of stocks. This distance was associated with the edge connecting the stocks, and it reflected the level at which they were correlated. We used a simple non-linear transformation <span class="math inline">\(d_{i, j}(t) = \sqrt{2(1 - \rho_{i,j}(t))}\)</span> to obtain distances with the property <span class="math inline">\(2 \geq d_{i,j} \geq 0\)</span>, forming a <span class="math inline">\(N \times N\)</span> symmetric distance matrix <span class="math inline">\(D(t)\)</span>.</p>
<div id="network-filtering-asset-graphs" class="section level3">
<h3><span class="header-section-number">6.2.1</span> Network Filtering: Asset Graphs</h3>
<p>We extract the <span class="math inline">\(N(N-1)/2\)</span> distinct distance elements from the upper triangular part of the distance matrix <span class="math inline">\(D(t)\)</span>, which were then sorted in an ascending order to form an ordered sequence <span class="math inline">\(d_1(t), d_2(t), \ldots, d_{N(N-1)/2}(t)\)</span>. Since we require the graph to be representative of the market, it is natural to build the network by including only the strongest connections. This is a network filtering procedure that has been successfully applied in the construction of  for the analyses of market structure . The number of edges to include is arbitrary, and we included those from the bottom quartile, which represented the 25% shortest edges in the graph (largest correlations), thus giving <span class="math inline">\(E(t) = \{d_1(t), d_2(t), \ldots, d_{\floor{N/4}}(t)\}\)</span>.</p>
<p>We denoted <span class="math inline">\(E^{F}(t)\)</span> as the set of edges constructed from the distance matrix derived from stock returns <span class="math inline">\(R(t)\)</span>. The financial network considered is <span class="math inline">\(G^{F} = ( V, E^{F} )\)</span>, where <span class="math inline">\(V\)</span> is the vertex set of stocks.</p>
</div>
<div id="network-filtering-mst" class="section level3">
<h3><span class="header-section-number">6.2.2</span> Network Filtering: MST</h3>
</div>
<div id="network-filtering-pmfg" class="section level3">
<h3><span class="header-section-number">6.2.3</span> Network Filtering: PMFG</h3>
</div>
</div>
<div id="applications" class="section level2">
<h2><span class="header-section-number">6.3</span> Applications</h2>
<div id="industry-taxonomy" class="section level3">
<h3><span class="header-section-number">6.3.1</span> Industry Taxonomy</h3>
</div>
<div id="portfolio-construction" class="section level3">
<h3><span class="header-section-number">6.3.2</span> Portfolio Construction</h3>

</div>
</div>
</div>



</div>
<h3>References</h3>
<div id="refs" class="references">
<div id="ref-aste2010correlation">
<p>Aste, Tomaso, W. Shaw, and Tiziana Di Matteo. 2010. “Correlation Structure and Dynamics in Volatile Markets.” <em>New Journal of Physics</em> 12 (8). IOP Publishing: 085009.</p>
</div>
<div id="ref-bardoscia2017pathways">
<p>Bardoscia, Marco, Stefano Battiston, Fabio Caccioli, and Guido Caldarelli. 2017. “Pathways Towards Instability in Financial Networks.” <em>Nature Communications</em> 8. Nature Publishing Group: 14416.</p>
</div>
<div id="ref-curme2015coupled">
<p>Curme, Chester, H Eugene Stanley, and Irena Vodenska. 2015. “Coupled Network Approach to Predictability of Financial Market Returns and News Sentiments.” <em>International Journal of Theoretical and Applied Finance</em> 18 (07). World Scientific Publishing Company: 1550043.</p>
</div>
<div id="ref-LI2018">
<p>Li, Yan, Xiong-Fei Jiang, Yue Tian, Sai-Ping Li, and Bo Zheng. 2018. “Portfolio Optimization Based on Network Topology.” <em>Physica A: Statistical Mechanics and Its Applications</em>. doi:<a href="https://doi.org/https://doi.org/10.1016/j.physa.2018.10.014">https://doi.org/10.1016/j.physa.2018.10.014</a>.</p>
</div>
<div id="ref-Mantegna1999">
<p>Mantegna, R. N. 1999. “Hierarchical Structure in Financial Markets.” <em>The European Physical Journal B - Condensed Matter and Complex Systems</em> 11 (1): 193–97. doi:<a href="https://doi.org/10.1007/s100510050929">10.1007/s100510050929</a>.</p>
</div>
<div id="ref-morales2012dynamical">
<p>Morales, Raffaello, Tiziana Di Matteo, Ruggero Gramatica, and Tomaso Aste. 2012. “Dynamical Generalized Hurst Exponent as a Tool to Monitor Unstable Periods in Financial Time Series.” <em>Physica A: Statistical Mechanics and Its Applications</em> 391 (11). North-Holland: 3180–9.</p>
</div>
<div id="ref-musmeci2014clustering">
<p>Musmeci, Nicoló, Tomaso Aste, and Tiziana di Matteo. 2014. “Clustering and Hierarchy of Financial Markets Data: Advantages of the Dbht.” <em>CoRR</em>.</p>
</div>
<div id="ref-pozzi2013spread">
<p>Pozzi, Francesco, Tiziana Di Matteo, and Tomaso Aste. 2013. “Spread of Risk Across Financial Markets: Better to Invest in the Peripheries.” <em>Scientific Reports</em> 3. Nature Publishing Group.</p>
</div>
<div id="ref-song2008analysis">
<p>Song, W.-M., T. Aste, and T. Di Matteo. 2008. “Analysis on Filtered Correlation Graph for Information Extraction.” <em>Statistical Mechanics of Molecular Biophysics</em>, 88.</p>
</div>
<div id="ref-song2012hierarchical">
<p>Song, W.-M., T. Di Matteo, and T. Aste. 2012. “Hierarchical Information Clustering by Means of Topologically Embedded Graphs.” <em>PLoS One</em> 7 (3). Public Library of Science: e31929.</p>
</div>
<div id="ref-Tumminello26072005">
<p>Tumminello, M., T. Aste, T. Di Matteo, and R. N. Mantegna. 2005. “A Tool for Filtering Information in Complex Systems.” <em>Proceedings of the National Academy of Sciences of the United States of America</em> 102 (30): 10421–6. doi:<a href="https://doi.org/10.1073/pnas.0500298102">10.1073/pnas.0500298102</a>.</p>
</div>
<div id="ref-Tumminello201040">
<p>Tumminello, Michele, Fabrizio Lillo, and Rosario N. Mantegna. 2010. “Correlation, Hierarchies, and Networks in Financial Markets.” <em>Journal of Economic Behavior &amp; Organization</em> 75 (1): 40–58. doi:<a href="https://doi.org/http://dx.doi.org/10.1016/j.jebo.2010.01.004">http://dx.doi.org/10.1016/j.jebo.2010.01.004</a>.</p>
</div>
<div id="ref-10.1371/journal.pone.0017994">
<p>Tumminello, Salvatore AND Lillo, Michele AND Miccichè. 2011. “Statistically Validated Networks in Bipartite Complex Systems.” <em>PLoS ONE</em> 6 (3). Public Library of Science: 1–11. doi:<a href="https://doi.org/10.1371/journal.pone.0017994">10.1371/journal.pone.0017994</a>.</p>
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